Simplify Trigonometric Expression

The task is to simplify the following trigonometric expression:

\( \frac{\sin A}{1-\cos A} + \frac{1-\cos A}{\sin A} \)

Step-by-Step Simplification

1. Identify the common denominator for the two fractions, which is \((1-cos A)sin A\).

2. Rewrite the expression with the common denominator:

   \( \frac{\sin^2 A}{(1-\cos A)\sin A} + \frac{(1-\cos A)^2}{\sin A (1-\cos A)} \)

3. Combine the numerators over the common denominator:

   \( \frac{\sin^2 A + (1-\cos A)^2}{(1-\cos A)\sin A} \)

4. Expand the squared term in the numerator: $(1-cos A)² = 1 - 2cos A + cos² A$.

   \( \frac{\sin^2 A + 1 - 2\cos A + \cos^2 A}{(1-\cos A)\sin A} \)

5. Use the Pythagorean identity $sin² A + cos² A = 1$.

   Substitute \(1\) for $sin² A + cos² A$:

   \( \frac{1 + (1 - 2\cos A)}{(1-\cos A)\sin A} \)

6. Simplify the numerator:

   \( \frac{2 - 2\cos A}{(1-\cos A)\sin A} \)

7. Factor out \(2\) from the numerator:

   \( \frac{2(1 - \cos A)}{(1-\cos A)\sin A} \)

8. Cancel the common factor \((1 - cos A)\), assuming \( A \neq 2n\pi \):

   \( \frac{2}{\sin A} \)

9. Apply the reciprocal identity \(cosec A = 1/sin A\):

   \( 2\cosec A \)

Final Result

The simplified expression is \(2\cosec A\).